Interesting information-theoretic quantity related to diatonicity

Something interesting I was thinking about, that I posted onto tuning earlier:

Mike wrote:
> In this case, consider that octaves are also the
> most common interval in the diatonic scale, and fifths the second-most
> common, whereas thirds carry the most "information" in a certain
> sense.
>
> This statement can be formulated quite precisely, in fact: consider
> the question, "what if I play a random type of fifth in the diatonic
> scale?" This implies a probability distribution: you have a 6/7 chance
> of playing a perfect fifth, and a 1/7 chance of playing a diminished
> fifth. The Shannon entropy of this distribution is obviously much
> lower than the entropy of the probability distribution of random types
> of thirds, which is maximal in the diatonic scale. And if you're
> working with a limited number of voices, you need to transmit as much
> information as possible, yes? So that'd be another possible route to
> explore this question, one which is related to the scales Bach
> generally uses - and one which might be worth considering, as omitting
> octaves and fifths as "less important" is common practice even if
> we're not talking about counterpoint at all.

Another good, related question is: which generic interval class gives
me the most information, on average, about where in the diatonic scale
I am? In other words, assume we hear a random dyad, and we want to
know what constellation of pitches forms the background scale, and
where our random dyad fits into that. Which generic class of dyad
maximizes the accuracy of my guess, on average? An equivalent question
is: if we call the lowest note of the dyad scale degree "1", which
mode is in the background? (Note that this equates things like C
ionian and D dorian, because we're not talking about "tonics" yet at
all.)

For any generic interval, this can be expressed using the mutual
information I(mode;majmin). This means, given an initial set of seven
modes with equal probability, how much entropy is reduced, on average,
by knowing that this generic interval is major or minor (or large or
small)?

What we really want is to ask the same question about the diatonic
scale in general. I'd like to say that the best way to formulate this
is to use the conditional mutual information
I(mode;majmin|genericinterval) - I'd like to assume that this is
right, but haven't checked it out enough yet to know for sure. What we
want to ask is, "how much entropy is reduced, on average and in
general, about which mode we're playing if we know the specific
interval size of some generic class of interval, assuming all generic
classes are played with equal probability?" If it's not the three-way
conditional mutual information then it's definitely one of the many
variants of it.

Assuming we find a suitable way to formulate this last question, we
should get the same value no matter what 7-note MOS we use. The real
magic is when we then consider the probability distribution of generic
intervals which maximizes the above result. This is like asking, which
distribution of generic intervals should I play to give the listener
as easy of an experience in figuring out where they are in the scale
as possible? As you might have guessed, I'm looking for something like
the channel capacity: the distribution of generic intervals producing
the supremum of conditional mutual information - but I can't find an
official definition of channel capacity which involves three variables
like this. Does anyone know what to use?

It's difficult to figure out what the results of something like this
would be given the lack of formalization on this last part, but I'd
predict that for most sensible things, as a general rule, the
probability for thirds and sixths should go up, and octaves and fifths
will go down. Of course, you have to formulate it in such a way that
the probability of everything but thirds and sixths doesn't go down to
zero...

Think about it! More to come.

-Mike